Regular spectrum of elements in topological algebras (Q1790519)

Let \(A\) be a topological algebra (with separately continuous multiplication). An element \(a \in A\) is said to be bounded if there is a positive number \(\lambda\) such that the sequence \(\{ (\frac{a}{\lambda})^n \}_n\) is bounded in \(A\). We denote by \(A_0\) the set of bounded elements of \(A\) and by \(\operatorname{Qinv}A\) the set of quasi-invertible elements of \(A\). In [J. Math. Pures Appl. (9) 33, 147--186 (1954; Zbl 0056.33601)], \textit{L. Waelbroeck} introduced the regular spectrum for unital commutative locally convex algebras and in [Proc. Lond. Math. Soc. (3) 15, 399--421 (1965; Zbl 0138.38202)], \textit{G. R. Allan} defined the regular spectrum for general unital locally convex algebras. In the paper under review, the author generalizes Allan's definition to the case of a general topological algebra \(A\), which is not necessarily unital. Let \(a \in A\). The regular spectrum \(\mathrm{sp}_A^r (a)\) is \[ \mathrm{sp}_A^r (a)= \mathrm{sp}_A(a) \cup \left\{\lambda \in \mathbb C \setminus \{0\}: \frac{a}{\lambda} \in \operatorname{Qinv}A \text{ and } \left(\frac{a}{\lambda}\right)^{-1}_q \not\in A_0\right\} \cup S , \] where \(S= \{ \infty\}\) if \(a \not\in A_0\) and \(S= \emptyset \) otherwise. The paper is devoted to the study of properties of the regular spectrum. In particular, the author discusses properties of the disolvent map \(D_a\), for \(a \in A\) in the case where \(A\) is a non-unital topological algebra, with \[ D_a : \mathbb C\setminus \mathrm{sp}_A(a) \to \mathbb C, \quad \lambda \mapsto \left(\frac{a}{\lambda}\right)^{-1}_q. \]